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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJul 3rd 2017
• (edited Jul 3rd 2017)

I have written out a detailed classical point-set proof that the fundamental group of the circle is the integers: here.

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeJul 3rd 2017

Another proof that might be nice to add at some point would use the groupoid version of the van Kampen theorem, applied to a standard good cover of $S^1$.

• CommentRowNumber3.
• CommentAuthorPeter Heinig
• CommentTimeJul 3rd 2017
• (edited Jul 3rd 2017)

Another proof that might be nice to add at some point would use the groupoid version […]

Coincidentally, there is a rather unusual treatment of $\pi(S^1)\cong \mathbb{Z}$, given by Tammo tom Dieck in his recent monograph “Algebraic Topology” (he emphasizes the groupoidal and functorial treatment, constructing a topological groupoid and an explicit isomorphism to $\Pi(S^1)$, and only in the end mentions the universal group as corollary about an automorphism group of an arbitrarily chosen point in the groupoid).

In the long run, as a result of the unique confluence of people and technology round here, the article fundamental group of the circle is the integers could become the most informed and modern treatment of this iconic theorem that the world has ever seen.

In the short run, a rendition of tom Dieck’s proof from his monograph (the proof is in Section 2.7 of the book), perhaps conceptually compared with Urs’s proof, could be a next step, possibly with some cross-referencing to fundamental groupoid, where $\pi(S^1)\cong\mathbb{Z}$ seems not yet been mentioned.

(edited to make it clearer; by “this theorem” I did not mean the van Kampen theorem, but the subject matter of fundamental group of the circle is the integers itself.)

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJul 3rd 2017

Todd, that’s a good idea. First we should type up an elementary proof of the van Kampen theorem itself.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJul 19th 2017

added a section “Consequences” (here), so far with the remark that iso-classes of finite covering spaces over the circle are labeled by conjugacy classes of symmetric groups, and the explicit example for three-sheeted coverings.